3. Random Number Generator

1. 3. Random Number Generator

2. The Roulette and Dice Mechanical random number generators

3. What is a Random Number? • Follow a definite distribution, usually uniform distribution • Uncorrelated • Unpredictable 1 0 3 7 4

4. Pseudo-Random Numbers • Truly random numbers can not be generated on a computer • Pseudo-random numbers follow a well-defined algorithm, thus predictable and repeatable • Have nearly all the properties of true random numbers

5. Linear Congruential Generator (LCG) • One of the earliest and also fastest algorithm: xn+1 = (axn + c ) mod m where 0 ≤ xn < m, m is the modulus, a is multiplier, c is increment. All of them are integers. Choice of a, c, m must be done with special care.

6. Choice of Parameters (ax + c) mod m

7. Short-Coming of LCG When (xn,xn+1) pairs are plotted for all n, a lattice structure is shown. xn+1 xn

8. Other Modern Generators • Mersenne Twister Extremely long period (219937-1), fast • Inversive Congruential Generator xn= a xn+1+ c mod m where m is a prime number Nonlinear, no lattice structure

9. Pick an Integer at Random • Suppose we want to select an integer j from 0 to N-1 with equal probability. This can be done with: j = N*x; where 0 x < 1 is uniformly distributed random number. For 2D lattice, we can also do this if we name the lattice site sequentially.

10. Pick j with Probability Pj • Since Sj Pj=1, we pick out j if x is in the corresponding interval. 0 P0 P1 x 1 P=p[0]; j = 0; x = drand64(); while(x>P) {++j; P+=p[j]};

11. Pei Lucheng’s Method • Use j = N*x to get an index; pick a final result based on the relative height. This is an O(1) algorithm. 1 4 2 1 2 3 4 5

12. Non-Uniformly Distributed Random Numbers • Let F(x) be the cumulative distribution function of a random variable x, then x can be generated from x = F-1(ξ) where ξ is uniformly distributed between 0 and 1, and F-1(x) is the inverse function of F(x).

13. Proof of the Inverse Method • The Mapping from x to ξ is one-to-one. • The probability for ξ between value ξ and dξ is 1·dξ, which is the same as the probability for x between value x and dx. Thus dξ = dF(x) = F’(x)dx = p(x)dx, since F-1 (ξ)=x, or ξ = F(x)

14. Example 1, Exponential Distribution • P(x) = exp(-x), x ≥ 0, then • So we generate x by x = -log(ξ) where ξ is a uniformly distributed random number.

15. Example 2, Gaussian distribution • Take 2D Gaussian distribution • Work in polar coordinates:

16. Box-Muller Method • The formula implies that the variable θ is distributed uniformly between 0 and 2π, ½r2 is exponentially distributed, we have ξ1 and ξ2 are two independent, uniformly distributed random numbers.